sampling a perfect sinusoid at Nyquist rate?

Vladimir Vassilevsky wrote:

> 
> 
> jjmai wrote:
> 
>> Let's say you have a perfect sine wave at frequency w.
>> According to Nyquist, in order to be able to recover the sine wave, you
>> need to have a sampling rate of at least 2w.  
> 
> 
> This is wrong.
> 
> If you know that the signal is a perfect sine wave, all you need is 3 
> samples to find the amplitude, the phase and the frequency.
> 
> VLV

Provided the sample interval is not an integer multiple of the period of 
the signal.

Mark wrote:

   ...

> well actually a pure sine wave has zero bandwidth so if you know the
> frequency, you don't need to sample it at all????

What if you don't? And what about amplitude and phase if you do?

Jerry
-- 
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Ron N. wrote:
> jjmai wrote:
> > Let's say you have a perfect sine wave at frequency w.
> > According to Nyquist, in order to be able to recover the sine wave, you
> > need to have a sampling rate of at least 2w.
> > So if you decide to sample at 2w, you end up with 2 samples for each cycle
> > of this sine wave.
>
> However the time it takes to recover the amplitude is proportional
> to the reciprocal of how close your sampling rate is to 2w.

can you expand upon that concept? derivation, assumptions, examples
etc. thanks

steve wrote:
> Ron N. wrote:
>> jjmai wrote:
>>> Let's say you have a perfect sine wave at frequency w.
>>> According to Nyquist, in order to be able to recover the sine wave, you
>>> need to have a sampling rate of at least 2w.
>>> So if you decide to sample at 2w, you end up with 2 samples for each cycle
>>> of this sine wave.
>> However the time it takes to recover the amplitude is proportional
>> to the reciprocal of how close your sampling rate is to 2w.
> 
> can you expand upon that concept? derivation, assumptions, examples
> etc. thanks

I don't have time now, but it's probably enough that the time to resolve 
(fs/2) - f is exactly the same as the time to resolve f to the same 
degree of accuracy. when f is very small, the time is long either way.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Sanctus  wrote:
> I doubt Nyquist said much at all - it was Shannon wasn't it?

Some say that this Russian Kotelnikov has precedence over both.
According to a discussion I had a couple of days ago, the French
mathematician Cauchy developed sampling theorey well before any of the
above three was born. It will probably turn out that Euler discovered
and stated the sampling theorem before him, and Archimedes before
Euler. It might well be that Newton and Leibniz also argued about it in
secret anagram communications. Oh, and don't forget Tesla - somewhere
in his papers, there _must_ be something as trivial as the sampling
theorem.

On 20 Dec 2006 23:18:51 -0800, "Andor" <andor.bariska@gmail.com>
wrote:

>
>
>Sanctus  wrote:
>> I doubt Nyquist said much at all - it was Shannon wasn't it?
>
>Some say that this Russian Kotelnikov has precedence over both.
>According to a discussion I had a couple of days ago, the French
>mathematician Cauchy developed sampling theorey well before any of the
>above three was born. It will probably turn out that Euler discovered
>and stated the sampling theorem before him, and Archimedes before
>Euler. It might well be that Newton and Leibniz also argued about it in
>secret anagram communications. Oh, and don't forget Tesla - somewhere
>in his papers, there _must_ be something as trivial as the sampling
>theorem.

Hi Andor,
  I thought Al Gore developed the Nyquist Sampling 
Theorem.

[-Rick-]

Rick wrote:
...
> >Some say that this Russian Kotelnikov has precedence over both.
> >According to a discussion I had a couple of days ago, the French
> >mathematician Cauchy developed sampling theorey well before any of the
> >above three was born. It will probably turn out that Euler discovered
> >and stated the sampling theorem before him, and Archimedes before
> >Euler. It might well be that Newton and Leibniz also argued about it in
> >secret anagram communications. Oh, and don't forget Tesla - somewhere
> >in his papers, there _must_ be something as trivial as the sampling
> >theorem.Hi Andor,
>   I thought Al Gore developed the Nyquist Sampling
> Theorem.

:-D